Stability and Bifurcation Analysis in a Class of Two-Neuron Networks with Resonant Bilinear Terms
A class of two-neuron networks with resonant bilinear terms is considered. The stability of the zero equilibrium and existence of Hopf bifurcation is studied. It is shown that the zero equilibrium is locally asymptotically stable when the time delay is small enough, while change of stability of the...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2011/697630 |
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| _version_ | 1832564018571640832 |
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| author | Changjin Xu Xiaofei He |
| author_facet | Changjin Xu Xiaofei He |
| author_sort | Changjin Xu |
| collection | DOAJ |
| description | A class of two-neuron networks with resonant bilinear terms is considered. The
stability of the zero equilibrium and existence of Hopf bifurcation is studied. It is shown that the zero
equilibrium is locally asymptotically stable when the time delay is small enough, while change of stability
of the zero equilibrium will cause a bifurcating periodic solution as the time delay passes through a
sequence of critical values. Some explicit formulae for determining the stability and the direction of the
Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal
form theory and center manifold theory. Finally, numerical simulations supporting the theoretical analysis
are carried out. |
| format | Article |
| id | doaj-art-e209a3f1117a46e9a4d293262670bf2c |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-e209a3f1117a46e9a4d293262670bf2c2025-02-03T01:12:04ZengWileyAbstract and Applied Analysis1085-33751687-04092011-01-01201110.1155/2011/697630697630Stability and Bifurcation Analysis in a Class of Two-Neuron Networks with Resonant Bilinear TermsChangjin Xu0Xiaofei He1Guizhou Key Laboratory of Economics System Simulation, Guizhou College of Finance and Economics, Guiyang 550004, ChinaDepartment of Mathematics, Zhangjiajie College of jishou University, Zhangjiajie 427000, ChinaA class of two-neuron networks with resonant bilinear terms is considered. The stability of the zero equilibrium and existence of Hopf bifurcation is studied. It is shown that the zero equilibrium is locally asymptotically stable when the time delay is small enough, while change of stability of the zero equilibrium will cause a bifurcating periodic solution as the time delay passes through a sequence of critical values. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Finally, numerical simulations supporting the theoretical analysis are carried out.http://dx.doi.org/10.1155/2011/697630 |
| spellingShingle | Changjin Xu Xiaofei He Stability and Bifurcation Analysis in a Class of Two-Neuron Networks with Resonant Bilinear Terms Abstract and Applied Analysis |
| title | Stability and Bifurcation Analysis in a Class of Two-Neuron Networks with Resonant Bilinear Terms |
| title_full | Stability and Bifurcation Analysis in a Class of Two-Neuron Networks with Resonant Bilinear Terms |
| title_fullStr | Stability and Bifurcation Analysis in a Class of Two-Neuron Networks with Resonant Bilinear Terms |
| title_full_unstemmed | Stability and Bifurcation Analysis in a Class of Two-Neuron Networks with Resonant Bilinear Terms |
| title_short | Stability and Bifurcation Analysis in a Class of Two-Neuron Networks with Resonant Bilinear Terms |
| title_sort | stability and bifurcation analysis in a class of two neuron networks with resonant bilinear terms |
| url | http://dx.doi.org/10.1155/2011/697630 |
| work_keys_str_mv | AT changjinxu stabilityandbifurcationanalysisinaclassoftwoneuronnetworkswithresonantbilinearterms AT xiaofeihe stabilityandbifurcationanalysisinaclassoftwoneuronnetworkswithresonantbilinearterms |